You'll notice right from the get go how similar both the rabbit and the bear are to Yertle the Turtle in the way they define themselves based on comparisons. Plus, they don't go through much of a journey other than getting more like this as the narrative wears on. So yeah, these guys are just like Yertles—but they probably run faster.
For the rabbit, it starts with "feeling mighty important that day" (1). Notice how we put "that day" in italics? That's because "that day" is pretty different from something like "always" or "as per usual." It's telling us, "Hey, this guy might be on top right now, but who knows where he'll be in a few pages, or even a few words?" The rabbit seems to know it, too, because he can only keep up his spirits by "[b]ragging," "boasting," "throwing out his chest," and just generally overcompensating for his insecurities (1).
But the thing about shouting out loud that you're better than everybody else is that you're going to upset someone else's vanity. Sure enough, the bear doesn't like this, and so the rabbit has no choice (so he thinks) but to keep on pumping himself up higher and higher. That is, of course, after a moment of doubt, when he wonders, '"Now what CAN I do?"' (6).
Of course, it's this pumping up that will ultimately be both his and the bear's undoing. He gets himself so high, he can't see that the real danger is from below. Remind you of Yertle… again? It should. And so should the rabbit's complete surprise at the end, without any real growth and repenting. How can he know that the real problem is inside when he's focused only on the outside?